Constraint satisfaction problems (CSPs) are a fundamental object of study in theoretical computer science. In quantum information theory, there are two natural analogues of CSPs, which both play important roles: local Hamiltonians and (our focus) non-local games. Non-local games originate from Bell’s pioneering 1964 paper, which showed how to test for quantum entanglement in a device with which we can interact only via classical inputs and outputs. In modern language, the tests developed by Bell are games: a referee presents two or more players with classical questions drawn from some distribution and demands answers from them. Each combination of question and answers receives some score and the players cooperate (but do not communicate) in order to maximize their expected score. These games are interesting because often the players can win the game with a higher probability if they share an entangled quantum state, so a high average score can certify the presence of quantum entanglement. Such tests are not only of scientific interest, but have had wide application to proof systems [7, 19], quantum key distribution [12, 1, 33], delegated computation , randomness generation  and elsewhere.
To be able to use a nonlocal game as a test for entanglement, it is essential to be able to approximately compute two quantities: the best possible expected score when the players share either classical correlations or entangled states, respectively called the “classical” and “quantum” (or “entangled”) values of the game, and denoted and . To understand these quantities, think of a game with players as inducing a constraint satisfaction problem with a -ary predicate. Each question in the game is mapped to a variable in the CSP, and each -tuple of questions and set of accepted responses (a “clause”) asked by the referee corresponds to a constraint. Classically, a simple convexity argument shows that the players can always stick to deterministic strategies, where each question is assigned a fixed answer; thus, is in fact identical to the value of the CSP. Hence, thanks to various dichotomy theorems, we have a good understanding of the difficulty of computing : in some cases, we know a algorithm, and for most others, we know it is -complete.
The quantum value is not as well understood. The main obstacle is that the set of entangled strategies is very rich: the “assignment” to each variable is no longer a value from a discrete set, but a linear operator over a Hilbert space of potentially unbounded dimension. As a result, we can say very little in terms of upper bounds on the complexity of computing . In fact, it is not known whether even a constant-factor (additive) approximation to is Turing-computable. For general games, the best we can say is that it is recursively enumerable: there is an algorithm, called the NPA or ncSoS hierarchy [22, 10], that in the limit of infinite time converges from above to the quantum value, but with no bound on the speed of convergence. On the hardness side, more is known, and what we know is grounds for pessimism: we have been able to show hardness results for approximating matching (e.g. ) and in some cases exceeding (e.g. ) the classical case by constructing special games that force entangled players to use particular strategies. Moreover, families of games have been found for which deciding whether is uncomputable . There are a few exceptions for which some positive results are known: for instance, the class of XOR games, in which the answers are bits and the payoff depends only on their XOR (for any given set of questions). In the classical case, these games are as hard as general games except in the “perfect completeness” regime: distinguishing satisfiability from satisfiability is -complete, but we can determine whether an XOR game is perfectly satisfiable in polynomial time using Gaussian elimination over . However, in the quantum case, it was shown by Tsirelon [5, 32] that for two-player XOR games, the lowest level of the ncSoS algorithm converges exactly to the quantum value, rendering it computable in polynomial time via semidefinite programming. (A similar technique was also applied to approximating the entangled value of unique games .) Yet these techniques seemed unlikely to generalize to three or more players: it is known that distinguishing satisfiability from for an entangled 3-player XOR game is -hard , and deciding the existence of perfect strategies for the closely-related family of linear systems games is uncomputable . For a summary of these results, see Table 1.
Another question which has been very fruitful in the study of classical CSPs is understanding the typical value of a random instance. Research in this direction draws significantly on insights from statistical mechanics and has proven that there exist sharp satisfiable/unsatisfiable thresholds for random
-SAT and related games (often using the equivalent constraint-satisfaction formulation). But these techniques do not carry over to the quantum case. For random classical games, a basic method is to look at the expected number of winning responses (the “first moment method”) or the variance (the “second moment method”) as we randomize the payoff function within some family such as random-SAT or random -XOR. This suffices, for example, to show that random -XOR games with variables and clauses are satisfiable with high probability if and only if . Since quantum strategies do not form a discrete (or even finite-dimensional) set, these methods are not possible. Nor is it obvious how to use more refined tools such as Shearer’s Lemma or the Lovasz Local Lemma, which address the question of when sets of overlapping constraints can be simultaneously satisfied. Indeed there are famous examples (such as the Magic Square game) of quantum “advantage” (i.e. the quantum value of a game is higher than the classical value) when there exist strategies for apparently contradictory constraints that succeed with probability 1. These suggest that the barriers to extending our classical intuition are not merely technical but reflect a genuinely different set of rules.
|Game type||Classical value||Entangled value|
or -complete depending on
|exact answer in [5, 32]|
|-quality approximation in |
in (this paper, Theorem 2.1)
recursively enumerable [22, 10]
Unique Games and the Unique Games Conjecture () are a generalization of 2-XOR games that are defined in . If the holds, then it is -complete to achieve any approximation ratio better than that known to be achieved by existing algorithms. For -XOR with it is -complete to beat the trivial approximation.
In this section we informally describe our main results, and then give precise theorem statements with links to proof sketches and full proofs later in the paper.
Our work introduces new techniques that let us make progress on the study of both worst-case complexity and random instances of XOR games with more than two players, in the regime where we are trying to decide whether . We think of a nonlocal game as a system of equations whose variables are linear operators, corresponding to the quantum measurements used by the players; a strategy is a solution to this system. Our main innovation is to consider a “dual” system of equations, whose solutions are objects that we call refutations. A refutation is a proof that the “primal” system of operator equations induced by the game is infeasible, and thus that . Surprisingly, for games that are symmetric under exchange of the players, we show that the dual system reduces to a set of linear equations over , which can be solved efficiently. This leads to our first result (Theorem 2.1), an algorithm for efficiently deciding whether for a symmetric -player XOR game, which brings the best known upper bound on this problem down from recursively enumerable [22, 10] to . See Table 1 for a summary of how our result fits in with known upper and lower bounds. Subsequently, by taking the dual of the dual, we are able to explicitly construct a set of quantum strategies (we call these Maximal Entanglement, Relative Phase, or MERP, strategies) that attain value 1 for all symmetric games with (Theorem 2.2). An explicit example shows that the symmetry assumption is indeed necessary for our algorithm to work: we exhibit a simple non-symmetric game called the 123 game, for which a simple, non-MERP strategy achieves , while our algorithm is unable to detect this (Theorem 2.3).
Our understanding of refutations and characterization of value-1 symmetric entangled games also lets us construct two specific families of games with interesting properties. The first, Capped GHZ (), is a family where ncSoS takes levels and time to detect that (Theorem 2.4), in contrast to our algorithm which runs in polynomial time. The second, Asymptotically Perfect Difference (), is an explicit, deterministic family of -XOR games with and classical value in the limit of large (Theorem 2.5). In comparison, there are randomized constructions of families of games whose bias ratio diverges for fixed as [24, 4]. However, known examples of these constructions involve both and , potentially precluding experimental distinguishability. To our knowledge this is the first construction of a family of XOR games that asymptotically achieves a perfect bias difference, , addressing one of the main open questions presented in .
For random instances, we analyze the dual system to show the existence of an unsatisfiable (i.e. ) phase as in the classical case (Theorem 2.6). We also relate our methods to the ncSoS hierarchy. For random instances, we show that in the unsatisfiable phase, a superlinear number of levels of ncSoS is necessary to certify that (Theorem 2.7).
The theorem statements of these results are as follows. Proof sketches are in Section 5.
There exists an algorithm that, given a -player symmetric XOR game with alphabet size and clauses, decides in time 111 Note that and do not scale independently for symmetric games. Any symmetric game may be specified by base clauses that are then symmetrized via at most permutations each, meaning . We could thus naively rewrite this runtime as to extract the dependence. Because the core information about the symmetric game is really only contained in the clauses, one might expect that it is possible to remove the factor of , and we hope to address this in a future work. whether or .
There exists a 6-player XOR game with alphabet size and clauses, for which but the algorithm of Theorem 2.1 cannot detect this.
Section 8.1. ∎
There exists a family of 3-XOR games with but for which the minimum refutation length scales exponentially in the number of clauses and alphabet size . For these games exponentially many levels of ncSoS are needed to witness that .
Section 8.2. ∎
There exists a family of -XOR games, parametrized by , for which and the classical value is bounded by
Section 8.3. ∎
For every , there exists a constant depending only on such that a random -XOR game with clauses has value with probability .
Section 9.2. ∎
For any constant , the minimum length refutation of a random 3-XOR game with queries on an alphabet of size has length at least
with probability (as ). Hence, either or levels of the ncSoS hierarchy are needed to witness that for such games.
Note that we can choose (with from Theorem 2.6) such that for large enough , typical random instances will have but ncSoS will require levels to detect this.
Section 9.3. ∎
3 Future Work
We see four main directions in which our characterization of non-local XOR games could be extended.
First, our linear algebraic characterization of games is incomplete: there exist games with for which a MERP strategy cannot achieve value 1. We expect a strengthing of the condition may allow us to extend our decidability algorithm to detect these games. An approach to this conjecture is described in Section 6 and a conjectured element of this set of games is given in Section 8.1. Understanding the structure of such games would give further intuition about the behavior of optimal XOR commuting-operator strategies, in particular strategies which may require more entanglement than the simple explicit strategies in Theorem 2.2.
Second, our results leave open the possibility that determining whether for nonsymmetric XOR games is outside or even undecidable. In the realm of Binary Constraint System (BCS) games,  shows that determining whether a general BCS game has perfect value is undecidable. The structural similarity between BCS games and XOR games suggests that perhaps some of the group theoretic techniques of that work could be applied to XOR games to arrive at a similar conclusion. An interesting class of games which may serve as an intermediate class between XOR and BCS games are “incomplete” XOR games in which there are players but queries can involve variables, effectively ignoring some players. Even for , Tsirelson’s semidefinite programming characterization of does not apply to incomplete XOR games, although in this case it is still easy to decide whether .
Thirdly, while in this work we have focused on computing the entangled game value , our methods may also be useful from the perspective of Bell inequalities, in which the quantity of interest is the maximal violation achievable by an entangled strategy. While this has conventionally been measured in terms of the bias ratio , the difference is an equally natural measure of violation, and we hope that our techniques will render it more amenable to analysis. Indeed, in addition to the construction of Asymptotically Perfect Difference games mentioned above, our results have the following simple consequence: for symmetric games with , our characterization of the optimal strategies (MERP) together with the Grothendieck-type inequality of  imply that the bias ratio and difference are both bounded by constants depending only on , and that for the difference, this constant is strictly less than one.
Finally, our results are almost entirely concerned with the question of whether or . However, we note that the class of strategies appearing in Theorem 2.2 include the optimal strategy for the CHSH game , but not for all XOR games . It is an interesting open question to find a natural characterization of games with for which MERP strategies are optimal. In this setting there are still many classical tools which we do not know how to extend to the classical case. As an example, consider overconstrained games in which there are many more constraints than variables and we choose the signs of those constraints randomly. In the classical case, the value is shown to be close to in Theorem 8.10 while in the quantum case we can only conclude that it is in Lemma 9.5.
Table 2 defines common notation used throughout the paper. This section is intended as a reference for the reader, while future sections provide more detailed technical definitions of these concepts.
|An XOR game. Consists of an indexed set of clauses.|
|The number of queries or clauses in an XOR game.|
|The question alphabet size or number of variables of an XOR game.|
|The number of players in an XOR game. We often use the phrase “-XOR game” to implicitly specify .|
|The -th query of a game. Written as a length-vector. The -th entry specifies the question sent to player .|
|The parity bit of the -th query in a game.|
|The -th clause of a game. Written as a vector of length that collectively represents the -th query and -th parity bit.|
|The classical value of XOR game .|
|The commuting-operator value of XOR game .|
|The , -valued game matrix for an XOR game. If then , otherwise .|
|The length-, -valued parity bit vector with entries .|
|The XOR game corresponding to game matrix and parity bit vector .|
|The measurement made by the -th player upon receiving question .|
|The strategy observable for player on question . Defined by the Hermitian operator .|
|The Hermitian operator representing the collective measurement made by the players of an XOR game upon receiving query . Defined by .|
|The shared state between the players of an XOR game.|
|A word. In general a -tuple with each element of the tuple a concatenation of questions (“letters”) drawn from . We frequently refer to the -th entry of by , and the specific letter at row and offset by .|
|The reversed form of a word.|
|Equivalence under neighboring pairs of letters canceling.|
|Equivalence under and parity-preserving permutations.|
5 Technical Overview
We begin by formally defining a -XOR game and its classical and quantum values.
Define a clause to be any -tuple consisting of a query and parity bit . In a -XOR game associated with a set of clauses , a verifier selects a clause uniformly at random from . Next, the question is sent to the -th player of the game, for all . The players then each send back a single output , and win the game if their outputs multiply to .
The key property of a game is its value – the maximum win probability achievable by players who cooperatively choose a strategy before the game starts, but cannot communicate while the game is being played. We distinguish various versions of the value by physical restrictions placed on the players.
For a given game , the classical value is the maximum win probability achievable by players sharing no entanglement.
The tensor-product value is the maximum win probability obtainable by players who share a quantum state but are restricted to making measurements on distinct factors of a tensor-product Hilbert space.
Finally, the commuting-operator value is the maximum win probability obtainable by players who may make any commuting measurements on a shared quantum state, not necessarily over a tensor-product Hilbert space. is often also referred to as the field-theoretic value of .
The commuting-operator value may differ from the tensor-product value of , and the question of whether it can differ from the closure of the set of values achievable by tensor product strategies remains open222And hard! For general games this question is known to be equivalent to Connes’ embedding conjecture .. In this paper, we focus primarily on a description of the commuting-operator value but in many cases can show that it coincides with the tensor-product value.
For the purpose of analyzing both the classical and commuting-operator value of -XOR games, we find it useful to define a linear algebraic representation for the game333There seems to exist an interesting parallel between this linear algebraic representation of an XOR game and the linear algebraic specification of a BCS game. While interesting, it is not explored in this work aside from its brief mention here and in Section 3.. The linear algebraic view represents queries as a matrix and parity bits as a vector. In doing so, it abstracts away from the specifics of labels and player/query indices to reveal the underlying game structure.
Given a -XOR game with queries and alphabet size , define the game matrix as an matrix describing query-player-question incidence. Specifically, can be written as a segmented matrix with distinct column blocks of size each, where the th column of block consists of a 1 in row if the th player receives question for query , and a 0 otherwise:
For such a game, define the length- parity bit vector by
An XOR game is completely specified by providing the game matrix and parity bit vector : . For example, the GHZ game  is defined by the clauses (here we use the labels for the questions instead of the typical ):
We translate the GHZ queries into and parity bits into by:
Many of our results apply to two special classes of XOR games: symmetric XOR games and random XOR games.
A symmetric -XOR game is an XOR game that additionally satisfies a clause symmetry property: for every clause in the game, the game must also contain all clauses where is a permutation of the questions in and the parity bit is unchanged.
A random -XOR game on clauses and variables is an XOR game
with the clauses chosen independently at random from a uniform
clauses chosen independently at random from a uniform distribution over.
We next introduce both classical and commuting-operator strategies and state claims regarding their value and constraints on when these strategies play perfectly given an XOR game. These claims are proved in Section 7.3. For any game, constructing a strategy and computing its value lower-bounds the value of the game. In the commuting-operator case, this is generally intractable and motivates the subsequent refutations picture.
5.1.1 Classical Strategies
For any game, the optimal classical strategy can be taken to be a deterministic assignment of answers. In the case of XOR games we will see that it is natural to view this assignment as a vector in .
A deterministic classical strategy dictates that player outputs when they receive question from the verifier. Note that valid outputs must satisfy
To exploit the linear algebraic picture, it is useful to define a length- classical strategy vector analogous to the parity bit vector. It is defined by the relation
Here the anticipates a generalization that we will see in the quantum case when we construct MERP strategies.
If the players play a game following strategy , the vector determines their output, i.e. query has answer . The value of classical strategy is
where again we have used an apparently unnecessary , anticipating a quantum generalization. We also treat and as equivalent here.
These observations lead to a well known procedure using Gaussian elimination to find a classical value-1 strategy or determine that no such strategy exists.
Define the classical constraint equations for game by
over . Equivalently,
When , on the other hand, there does not exist an efficient algorithm for finding optimal classical strategies (assuming ) .
5.1.2 Commuting-Operator Strategies
Consider a -XOR game with variables. For each , let the Positive-Operator Valued Measure (POVM) give the -th player’s commuting-operator strategy upon receiving question from the verifier. These POVMs act on some shared state , and different players’ POVM elements commute due to the no-communication requirement on the players.
Using the Naimark dilation theorem, we can restrict our players’ strategies to be Projection-Valued Measures (PVMs). We make this restriction for the remainder of the paper. This allows us to define the following observables.
Given a strategy , define the strategy observable
Since is a PVM, is a Hermitian operator. Indeed commuting-operator strategies are equivalent to imposing the constraints for
|(operators held by distinct players commute)||(13a)|
|(square identity, analogous to (8))||(13b)|
The condition for commuting-operator strategies to achieve value 1 is the following generalization of (11).
For a -XOR game , define the commuting-operator constraint equations:
These equations stipulate that applying the strategy observables for a given question to the shared state produces the correct output for that question.
While there is an efficient algorithm to solve the classical constraint equations, no such algorithm is known to exist for the commuting-operator constraint equations. This difficulty forces us to consider alternative techniques for characterizing the commuting-operator value of XOR games.
In addition to lower bounding the value of a game by constructing strategies for it, we can also upper bound a game’s value by showing no high-value strategy can exist. In particular, we construct proofs that a game cannot have value 1, which we call refutations. Classically, refutations are well understood, and emerge naturally from the dual to the classical constraint equations.
Define a classical refutation as any vector satisfying the equations dual to (11),
where once again the algebra is over .
The proof is standard but because dualities like Fact 5.15 play a major role in our paper, we review it here.
By the definition of and , we have . The rank-nullity theorem implies that , meaning that in fact
Another way to view as a refutation is by interpreting it as the indicator vector of a subset of clauses. Recall from (12) that clause corresponds to the equation over the variables . If satisfies (15) then multiplying the equations corresponding to clauses with yields
In this paper we consider the commuting-operator analogue of classical refutations. We would like to construct a dual to (14), meaning a characterization of certificates for the unsatisfiability of (14). As there is no analogue to the linear algebraic methods used in the classical case, we will instead attempt to generalize (18).
Cleve and Mittal  make use of a noncommutative generalization of (18), which they call the substitution method, to exhibit refutations of some Binary Constraint System games. We will use a similar method for XOR games in which we multiply together constraints of the form (14) to obtain a contradiction. Our contribution will be to give a simple characterization of when such refutations exist in the case of symmetric -XOR games and in some cases, random asymmetric 3-XOR games. Indeed, our characterization will resemble the classical dual equations (15) although the route by which we obtain it is quite different.
To explain this in more detail, we introduce some definitions.
Let and be two operators formed from products of strategy observables. We say is equivalent to , written , if is an identity for all strategy observables satisfying (13).
Let be some -XOR game with clauses. A refutation for is defined to be a sequence satisfying
Refutations certify that , analogous to the way that classical refutations certify that . In Theorem 6.1, we show that in fact any game with has a refutation. The proof of this fact relies on a connection between refutations and the ncSoS hierarchy analogous to a connection made by Grigoriev  between classical refutations and the SoS hierarchy.
It is not obvious that one can find refutations more easily than one can find strategies. However, we next establish a necessary condition for a game to admit a refutation, and thus an easily-identified subclass of XOR games that certainly do not have a refutation meaning they have .
5.3 Games with no Parity-Permuted Refutations (noPREF Games)
noPREF games are a subclass of entangled XOR games for which it is easy to show no refutation can exist. To motivate their construction and prove some properties about them, we must first redefine refutations from a combinatorial perspective. A more complete treatment of these ideas is given in Section 6.
Definition 5.18 (Combinatorial Construction of Refutations, informal).
For a -XOR game , consider the combinatorial version of the query 444 We overload the notation here to indicate both the definitional and combinatorial version of a query, with the relevant meaning clear from context. to be a vector with coordinates (the player indices) with letter at coordinate . Define the set of words contained in to be all vectors formed by concatenating the queries of coordinate-wise (by player). The sign of a word contained in
is defined as
We will refer to each coordinate of the word as a wire. The identity under the concatenating action is the word that is blank on every wire.
Define an equivalence relation generated by all wire-by-wire permutations of the following base relations (in this setting the product of two vectors indicates their coordinate-wise concatenation).
(Repeated elements cancel) :
(Elements on different wires commute) :
A refutation is defined to be a sequence for which
We claim that this definition of a refutation is equivalent to the one given in Section 5.2. Intuitively, such a construction explicitly manipulates the operator identities required by each clause of in a way that exploits the operator requirements of (13) to produce a refutation as in Definition 5.17. We prove this fact in Section 6. We next motivate the noPREF class of games by making the following key observation.
All elements contained in queries at even positions in a refutation must cancel with queries at odd positions.
To exploit this observation, we find it useful to define a broader equivalence relation that allows for a parity-preserving permutation on each wire before canceling and commuting letters.
Definition 5.20 (Informal).
We say -XOR word is parity-permuted equivalent to —denoted —if where some permutations of the even positions and odd positions on each wire of can produce .
Since this is just a broadening of the equivalence given in Definition 5.18, . With this equivalence relation in hand, we can state a necessary condition for the existence of a refutation of a game .
A game contains a Parity-Permuted Refutation (PREF) if the game contains a word which is with sign .
The set of PREF Games are the set of XOR games that contain PREFs. The set of noPREF Games are the set of XOR games that do not.
Theorem 5.22 (Necessary condition for refutation).
If a game admits a refutation, it contains a PREF.
This follows essentially immediately from the observation that implies . ∎
Every noPREF game has commuting-operator value 1.
The significance of noPREF games is made clear by the two following theorems. For both, a short proof sketch is given, while the full proofs are delegated to Section 6.
Theorem 5.24 (Informal).
There exists a poly-time algorithm that decides membership in the set of noPREF games.
The key observation here is that a game contains a PREF if and only if there is a solution to the set of equations
such that their elements are placed in odd and even positions, respectively. The reverse direction requires a technical lemma relating the even and odd clauses of a PREF. Then standard techniques for solving linear Diophantine equations complete the proof. ∎
Theorem 5.25 (Informal).
The noPREF characterization is complete for symmetric games. That is, every value 1 symmetric game is in the noPREF set.
We use the structure of symmetric games to construct shuffle gadgets – short words that move letters from one wire to another when they are appended onto an existing word. We then show shuffle gadgets are sufficient to construct a refutation given a PREF contained in the game. This shows that containing a PREF is both necessary and sufficient for a symmetric game to have a refutation. Then a symmetric game is either in the set of noPREF games or has value . ∎
Theorems 5.24 and 5.25 together show that the class of symmetric value 1 games has a poly-time deterministic algorithm, while previously the question of whether such games took value 1 was not known to be decidable. This progress is due to the noPREF characterization of games.
Given that noPREF games form a large class of value 1 games, it is reasonable to try to construct a commuting-operator strategy to play them. We can ask if there exists a strategy dual to the PREF criteria, similar to what we have described in the classical and commuting-operator cases. In particular, we ask if a game not satisfying (23) and (24) guarantees existence of a solution to the constraint equations indicating some simple family of strategies can achieve value 1 for .
Somewhat miraculously, the answer to this question turns out to be yes. We proceed by first defining this class of strategies, then showing that their constraint equations are dual to the PREF criteria for any game.
5.4 Maximal Entanglement, Relative Phase (MERP) Strategies
We introduce a family of “Maximal Entanglement, Relative Phase” (MERP) strategies: a useful subfamily of the set of tensor-product (and thus commuting-operator) strategies. MERP strategies are a generalization of the GHZ strategy to arbitrary games. Crucially, determining whether a MERP strategy achieves value 1 for a game, and if so a construction for such a strategy, can be described in time polynomial in , , and .666 For symmetric games, , so in this case one can decide MERP value 1 and describe a strategy in time polynomial in and .
Furthermore, the conditions for a MERP strategy to achieve value 1 are dual to the PREF condition for a game, meaning MERP achieves value 1 on any noPREF game. In particular, this means MERP strategies achieve tensor-product value 1 on any symmetric XOR game with (Theorem 5.25) as well as on a family of non-symmetric games (APD games, Section 8.3) with and classical value .
We begin with the definition of a MERP strategy for a game .
Definition 5.26 (Merp).
Given a -XOR game with m clauses, a MERP strategy for is a tensor-product strategy in which:
The players share the maximally entangled state
with player having access to the -th qubit of the state.
Upon receiving question from the verifier, player rotates his qubit by an angle about the axis, then measures his qubit in the basis and sends his observed outcome to the verifier.
Explicitly, we define the states
and pick strategy observables
There exists a useful parallel between MERP strategies and classical strategies, which we summarize below. Almost identically to the classical value (10),
Let the length- MERP strategy vector for a given MERP strategy be defined by
The value achieved by that MERP strategy on game is:
Explicit calculation. Done in full in Section 7.2. ∎
Claim 5.27 allows us to write down the constraint equations for MERP strategies to achieve .
Define the MERP constraint equations for game by
(We could have equivalently required to be in . This is because have integer entries and so any real solution to (31) will also be rational.)
A MERP strategy achieves on a game iff its MERP constraint equations have a solution. A solution corresponds to the MERP strategy in which player uses .
Intuitively, MERP provides an explicit construction allowing players to return an arbitrary phase on each input, rather than the classical or . The MERP constraint equations then ensure that for each question the returned phases sum to up to multiples of . For any game, Claim 5.29 allows us to efficiently determine whether some MERP strategy achieves value 1 via Gaussian elimination over . We often refer to this optimal MERP strategy777Despite the language, we do not wish to suggest that there is a single optimal MERP strategy. Instead one should imagine some convention being used to specify a unique MERP strategy from the set of optimal ones. for a game as simply the MERP strategy for .
5.5 MERP - PREF Duality
The set of games for which MERP achieves value 1 is exactly the set noPREF. As in the classical and commuting-operator cases, the MERP constraint equations (31) are dual to the PREF conditions:
For any game , either there exists a PREF specification, or a MERP strategy with value 1.
Because of Theorem 5.30 we also refer to a PREF specification as a MERP refutation.
Figure 1 summarizes the extensions of the classical duality relations presented in this paper. The general quantum duality provides a complex but complete description of games with . The PREF conditions are efficient to compute, but are only necessary conditions for constructing commuting-operator refutations, and thus the dual, MERP value 1, holds true for only a subset of all games. We can make a stronger statement about symmetric games: PREFs are both necessary and sufficient for a symmetric game to have a refutation, so the duality ensures MERP achieves value 1 for all symmetric games with .
Finally we can use our main results to analyze some particular families of games and partially characterize the XOR game landscape.
In the regime, we construct a family of games that generalize the GHZ game, termed the Asymptotically Perfect Difference (APD) family. Members are parameterized by scale , with the -th member having players, and reproducing GHZ. The APD family is contained in the noPREF set () and has perfect difference in the asymptotic limit,
This demonstrates that XOR games include a subset for which (at least asymptotically) the best classical strategy is no better than random while a tensor-product strategy (MERP) can play perfectly. Details of this construction are given in Section 8.3. We also give, in Section 8.1, the construction for a (nonsymmetric) game for which but which falls outside the noPREF set, which shows the incompleteness of the PREF criteria.
To study the regime, we consider the behavior of randomly generated XOR games with a large number of clauses. We prove Theorem 2.6 by explicitly constructing a refutation for such games using insights developed in previous sections. Interestingly, we also show such games have a minimal length refutation that scales like , which implies that it takes the ncSoS algorithm superexponential time to show that these games have (Lemma 6.4 and Theorem 2.7). These results can be seen as quantum analogues of Grigoriev’s  integrality gap instances for classical XOR games. Finally, we try to push the potentially superexponential runtime of ncSoS to its extremes. We demonstrate a family of symmetric games, called the Capped GHZ family, that provably have , but have minimum refutation length exponential in the number of clauses (Section 8.2). For games in this family the ncSoS algorithm requires time doubly exponential to prove that their commuting-operator value is while the noPREF criterion can be used to conclude this fact in polynomial time.
Refutations are a powerful tool for differentiating between XOR games with perfect commuting-operator strategies () and those with bounded away from 1. In Section 6.1, we prove Theorem 6.1 and Theorem 6.2, relating refutations to the commuting-operator value of XOR games:
An XOR game has commuting-operator value if and only if it admits no refutations.
Let be an XOR game consisting of queries, with yielding a length- refutation. The commuting-operator value of the game is bounded above by
We previously introduced the notion of the combinatorial view of refutations (Definition 5.18) and containing a PREF as a necessary condition for a game to have a refutation (Corollary 5.23). Section 6.2 presents the combinatorial view in more detail, and proves that a PREF specification and existence of a particular set of “shift gadgets” is a sufficient condition for a refutation to exist. Finally, Section 6.3 demonstrates that for symmetric XOR games, all desired “shift gadgets” are automatically available, meaning that a refutation exists if and only if a PREF specification exists, thus providing an efficient technique to decide whether any symmetric XOR game has perfect commuting-operator value.
6.1 Upper Bound on Value
We begin by proving Theorem 6.1. The main tool we use is the non-commuting Sum of Squares (ncSoS) hierarchy, also known as the NPA hierarchy [22, 10]. Given a game , each level in the ncSoS hierarchy is a semidefinite program depending on whose solution gives an upper bound on the value ; higher levels correspond to larger semidefinite programs and tighter upper bounds. While we refer the reader to the references cited above for a full description, we include here a definition of the key object used in constructing the hierarchy: the pseudoexpectation operator.
Given an XOR game , a degree- pseudoexpectation operator or pseudodistribution is a linear function that maps formal polynomials of degree at most over the strategy observables to complex numbers. A pseudoexpectation is valid if
for all polynomials of degree at most , ,
for all polynomials with and indices and ,
for all polynomials with and indices and ,
Intuitively speaking, these requirements state that any algebraic manipulations allowed by (13) are also allowed under the pseudoexpectation, as long as they never result in a polynomial of degree greater than . We further say that a pseudoexpectation satisfies a clause if for all polynomials with degrees summing to , .
The full ncSoS algorithm involves optimizing over all valid pseudoexpectation operators that satisfy clauses in the game; it can be shown that this optimization reduces to a semidefinite program in matrices whose dimension is the number of monomials of degree at most in the observables . In the special case of determining whether the game value is , it reduces to checking for the existence of such a pseudoexpectation operator.
In , Grigoriev showed a connection between refutations of classical games and pseudodistributions which appear to satisfy all clauses of a classical XOR game. In our analysis, we will adapt some of these arguments to the quantum setting. In particular, Lemma 6.4 gives a quantum analogue of Grigoriev’s central insight that, in the special case of deciding whether the game value is , the sum-of-squares hierarchy reduces to checking for the existence of a refutation.
In addition to being key to the proof of Theorem 6.1, Lemma 6.4 also gives a bound on the time it takes the ncSoS algorithm to show a XOR game has value in terms of the minimum length refutation admitted by the game.
For any -XOR game with no refutation of length there exists a degree- pseudodistribution whose pseudoexpectation satisfies every clause in . Consequently, it takes time at least for the ncSoS algorithm to prove .
To construct this pseudodistribution, we follow a procedure of Grigoriev . For each clause , define
and for any word888 In this context, we borrow this terminology from the combinatorial picture to indicate any product of strategy observables. which can be obtained as a product of queries,
define the pseudoexpectation of to be the product of the parity bits associated with each query in the operator construction:
We need to argue that this prescription is well-defined, i.e. that (37) and (38) never assign two different values to the same . Suppose to the contrary that with but that . Since (38) can only take on the values we have . Also each is Hermitian, so
This constructs a refutation of length , contradicting our hypothesis that no such refutation exists. We conclude that is well-defined for the choices of resulting from (37).
For all other words (i.e. those that cannot be obtained as products of queries or have length ), set their pseudoexpectation to . Finally, extend the definition by linearity to sums and scalar multiples of operator products.
Moreover, induces an equivalence relation on words: we say that words if